Random walk speed is a proper function on Teichmüller space

Aitor Azemar; Vaibhav Gadre; Sébastien Gouëzel; Thomas Haettel; Pablo Lessa; Caglar Uyanik

doi:10.3934/jmd.2023022

Article Contents

2023, Volume 19: 815-832. Doi: 10.3934/jmd.2023022

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Random walk speed is a proper function on Teichmüller space

1.
School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, UK
2.
IRMAR, CNRS UMR 6625, Université de Rennes, 35042 Rennes, France
3.
IMAG, Université de Montpellier, CNRS, 34095 Montpellier, France
4.
IRL 3457, CRM-CNRS, Université de Montréal, Montréal H3C 3J7, Canada
5.
IMERL, Universidad de la República, Montevideo, 11200, Uruguay
6.
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

Received: January 20, 2023

Revised: June 11, 2023

Published online: October 13, 2023

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Abstract

Consider a closed surface $ M $ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $ M $ with a finite first moment. Corresponding to each point in the Teichmüller space of $ M $, there is an associated random walk on the hyperbolic plane. We show that the speed of this random walk is a proper function on the Teichmüller space of $ M $, and we relate the growth of the speed to the Teichmüller distance to a basepoint. One key argument is an adaptation of Gouëzel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.

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Mathematics Subject Classification: Primary: 05C81, 20F67; Secondary: 60B20, 30F45, 20E08.

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